Question

$\int e^x\left\{\frac{1+\sin x\cos x}{{\cos }^{2}x}\right\}dx=$

• A. $e^x\cos x+c$
• B. $e^x\sec x\tan x+c$
• C. $e^x\tan x+c$
• D. $e^x{\cos }^{2}x-1+c$

Answer 1

The correct option is C $e^x\tan x+c$

We have $\int e^x\left(\frac{1+sinx.cosx}{co{s}^{2}x}\right)dx$ $=\int e^x(se{c}^{2}x+tanx)dx$
$=\int e^{x}tanxdx+\int e^{x}se{c}^{2}dx$
Applying ILATE rule to the first integral.
$=e^{x}tanx-\int se{c}^{2}x{e}^x.dx+\int e^{x}se{c}^{2}dx$
$=e^{x}tanx+C$.